Unformatted text preview: about the other players’ actions is not assumed to be correct,
but rather, simply constrained by rationality.
Deﬁnition
A belief of player i about the other players’ actions is a probability
measure σ−i ∈ ∏j �=i Σj (recall that Σj denotes the set of probability
measures over Sj , the set of actions of player j ). 18 Game Theory: Lecture 3 Mixed Strategy Equilibrium Rationality
Rationality imposes two requirements on strategic behavior:
(1) Players maximize with respect to some beliefs about opponent’s
behavior (i.e., they are rational).
(2) Beliefs have to be consistent with other players being rational, and
being aware of each other’s rationality, and so on (but they need not
be correct).
Rational player i plays a best response to some belief σ−i .
Since i thinks j is rational, he must be able to rationalize σ−i by thinking
every action of j with σ−i (sj ) > 0 must be a best response to some belief j
has.
.
.
.
Leads to an inﬁnite regress: “I am playing strategy σ1 because I think player
2 is using σ2 , which is a reasonable belief because I would play it if I were
�
player 2 and I thought player 1 was using σ1 , which is a reasonable thing to
� is a b est response to σ � , . . ..
expect for player 2 because σ1
2
19 Game Theory: Lecture 3 Mixed Strategy Equilibrium Example
Consider the game (from [Bernheim 84]),
a1
a2
a3
a4 b1
0, 7
5, 2
7, 0
0, 0 b2
2, 5
3, 3
2, 5
0, −2 b3
7, 0
5, 2
0, 7
0, 0 b4
0, 1
0, 1
0, 1
10, −1 There is a unique Nash equilibrium (a2 , b2 ) in this game, i.e., the strategies a2
and b2 rationalize each other. Moreover, the strategies a1 , a3 , b1 , b3 can also be
rationalized:
Row will play a1 if Column plays b3 .
Column will play b3 if Row plays a3 .
Row will play a3 if Column plays b1 .
Column will play b1 if Row plays a1 .
However b4 cannot be rationalized, and since no rational player will play b4 , a4
can not be rationalized.
20 Game Theory: Lecture 3 Mixed Strategy Equilibrium NeverBest Response Strategies
Example
Consider the following game: Q
X
F Q
4, 2
1, 1
3, 0 F
0, 3
1, 0
2, 2 It can be seen that F can be rationalized.
If player 1 believes that player 2 will play F, then playing F is rational
for player 1, etc.
However, playing X is never a best response, regardless of what strategy is
chosen by the other player, since playing F always results in better payoﬀs.
A strictly dominated strategy will never be a best response, regardless of a
player’s beliefs about the other players’ actions.
21 Game Theory: Lecture 3 Mixed Strategy Equilibrium NeverBest Response and S...
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 Spring '10
 AsuOzdaglar
 Game Theory, mixed strategy

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